On combinatorial aspects of modules over commutative rings

TL;DR

This paper explores the deep connections between combinatorial graph structures and algebraic modules over commutative rings, providing characterizations and equivalences that enhance understanding of module properties.

Contribution

It introduces a novel framework linking graph theory with module theory, characterizing finite abelian groups and analyzing essential ideals and annihilating graphs.

Findings

A combinatorial object determines an algebraic module.

Characterization of all finite abelian groups.

Analysis of isomorphisms of annihilating graphs.

Abstract

Let R be a commutative ring with unity, M be an unitary R-module and {\Gamma} be a simple graph. This research article is an interplay of combinatorial and algebraic properties of M . We show a combinatorial object completely determines an algebraic object and characterise all finite abelian groups. We discuss the correspondence between essential ideals of R, submodules of M and vertices of graphs arising from M . We examine various types of equivalence relations on objects of M. We study essential ideals corresponding to elements of an object over hereditary and regular rings. Further, we study isomorphism of annihilating graphs arising from M and tensor product.